In English and some other languages, the word "crusade" has been used occasionally, especially in the past, to mean "any concerted effort" by a group, with only a vague reference to the actual Crusades and not necessarily religious connotation.
Did the Soviets ever refer to "crusades" in that way, like "crusade for the workers" or "crusade for the revolution", or would it be out of character?
In Russian "crusade" is even more specific - крестовый поход, the cross expedition/campaign.
Since the "cross" part is more readily recognizable in Russian, any positive use of the phrase was absolutely out question in the militarily atheistic USSR, which officially viewed crusades as evil.
Moreover, the phrase had a distinct politically negative connotation ever since Pius XI's 1930 letter to Cardinal Pompili https://w2.vatican.va/content/pius-xi/it/letters/documents/hf_p-xi_lett_19300202_ci-commuovono.html criticizing the religious persecution in the USSR. He supported the people joining in on a "crusade of prayers".
The Soviet propaganda omitted the "prayers" part and presented the letter as an attempt to start a "crusade against the USSR" (Bukharin got the agitprop ball rolling with an article in Pravda). This phrase (in various versions, e.g. "crusade against communism") remained a propaganda cliche for decades afterwards (source: https://cyberleninka.ru/article/v/holodnaya-voina-i-ideologicheskaya-borba-na-religioznom-fronte-o-nekotoryh-modelyah-sovetskoy-propagandy ).
In general, all actions by the West ("Anglo-American imperialism") during the Cold War perceived as hostile could be said to be a part of this "crusade".
Thus the positive use of the phrase was practically impossible for the two reasons described above.
Also, the crusades were a Western-Church phenomenon, not a part of the Russian history, psychologically - something that "they" do and thus hard to use as a catchphrase for something that "we" do.